MATH1014_integration_III_Fall_2019_20.pdf - Techniques of Integration Trigonometric substitutions(Mainly based on Stewart Chapter 7 \u00a77.3 Edmund Chiang

# MATH1014_integration_III_Fall_2019_20.pdf - Techniques of...

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Techniques of Integration: Trigonometric substitutions (Mainly based on Stewart: Chapter 7: § 7.3) Edmund Chiang MATH1014 September 25, 2019 1 Pythagoras theorem Trigonometric functions are about description of circles. Since trigonometric ratios are in- dependent of the sizes of the circles (right-angled triangles) so we normalise the radius of circle to unit radius as shown in the figure below. Figure 1: (Different line segments represent different trigonometric ratios) 1
Identities: The above figure suggests the following identities: sin 2 θ + cos 2 θ = 1 , tan 2 θ + 1 = sec 2 θ, 1 + cot 2 θ = csc 2 θ Writing cos θ = x , similar triangles consideration yields sin θ = p 1 - x 2 cot θ 1 = cos θ sin θ = x 1 - x 2 , sec θ 1 = 1 sin θ = 1 1 - x 2 csc θ 1 = 1 cos θ = 1 x , tan θ 1 = sin θ cos θ = 1 - x 2 x . If, however, we have circles whose radius a 6 = 1 , then we simply ”scale up/down” the above identities by a factor of a . 2 Trigonometric substitutions Our main task is to integrate expressions involving p a 2 - x 2 dx, x = a sin θ p
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